Aspects of Riemannian geometry in quantum field theories

Aspects of Riemannian Geometry in Quantum

Field Theories

by

Ricardo Schiappa

Licenciado in Physics,

Instituto Superior T6cnico (Lisbon, Portugal), June 1994

Submitted to the Department of Physics

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY IN PHYSICS

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 1999

©

Massachusetts Institute of Technology 1999. All rights reserved.

Author...

...

...

...

bepartment of Physics

April 30, 1999

C ertified by ...

...

-

.

...

Jeffrey Goldstone

Cecil & Ida Green Professor of Physics

Thesis Supervisor

A ccepted by...

.

..

...

Tho&mas J.Peytak

Professor, Associate Department Head for Education

MASSACHUSETTS INSTITUTE I CNOLOGY

Aspects of Riemannian Geometry in Quantum Field

Theories

by

Ricardo Schiappa

Submitted to the Department of Physics on April 30, 1999, in partial fulfillment of the

requirements for the degree of

DOCTOR OF PHILOSOPHY IN PHYSICS

Abstract

In this thesis we study in detail several situations where the areas of Riemannian geometry and quantum field theory come together. This study is carried out in three distinct situations. In the first part we show how to introduce new local gauge invariant variables for V = 1 supersymmetric Yang-Mills theory, explicitly

parame-terizing the physical Hilbert space of the theory. We show that these gauge invariant variables have a geometrical interpretation, and that they can be constructed such that the emergent geometry is that of

AV

= 1 supergravity: a Riemannian geome-try with vector-spinor generated torsion. In the second part we study bosonic and supersymmetric sigma models, investigating to what extent their geometrical target space properties are encoded in the T-duality symmetry they possess. Starting from the consistency requirement between T-duality symmetry and renormalization group flows, we find the two-loop metric beta function for a d = 2 bosonic sigma model on a generic, torsionless background. We then consider target space duality transforma-tions for heterotic sigma models and strings away from renormalization group fixed points. By imposing the consistency requirements between the T-duality symme-try and renormalization group flows, the one loop gauge beta function is uniquely determined. The issue of heterotic anomalies and their cancelation is addressed from this duality constraining viewpoint, providing new insight and mechanisms of anomaly cancelation. In the third part we compute a radiative contribution to an anomalous correlation function of one axial current and two energy-momentum ten-sors,

(A,,(z)T,,,(y)Tp,(x)),

corresponding to a contribution to the gravitational axial anomaly in the massless Abelian Higgs model. In all three situations there is a rich interplay between geometry and field theory.

Thesis Supervisor: Jeffrey Goldstone

Acknowledgments

First and foremost I would like to thank my advisor, Kenneth Johnson, for all the teaching, discussions and suggestions, that molded a great deal of the present thesis. I would like to thank Kenneth Johnson, Daniel Freedman and Washington Tay-lor IV for advising me on specific research projects through many discussions and suggestions that taught me a great deal of physics.

I would like to thank Rui Dilio, Peter Haagensen, Kasper Olsen, Jiannis Pachos, Orfeu Bertolami, Washington Taylor IV, Lorenzo Cornalba and Joio Nunes for very stimulating collaboration in several research projects and/or papers.

I would like to thank Kenneth Johnson, Peter Haagensen, Joio Nunes, Poul Damgaard, Daniela Zanon, S. Belluci, A.A. Tseytlin, Daniel Freedman, Joshua Er-lich and Roman Jackiw for reading and commenting on my papers, for suggestions or comments about them, and for discussions that helped shape those papers.

I would like to thank Gustavo Granja, Daniel Chan, Joio Nunes, Jos6 Mourio, Rui Dilio, Kenneth Johnson, Pascal Letourneau, Peter Haagensen, Kasper Olsen, Joio Correia, Daniel Freedman, S. Belluci, Jiannis Pachos, Joshua Erlich, Pedro Fon-seca, Orfeu Bertolami, Washington Taylor IV, Lorenzo Cornalba and Alec Matusis, amongst many other people, for several discussions on physics and mathematics.

I would also like to thank my thesis committee, Jeffrey Goldstone, Daniel Freed-man and Mehran Kardar, for reading of the present thesis.

Finally, I would like to thank the several sources of financial aid I enjoyed through-out these years: Praxis XXI grant BD-3372/94 (Portugal), Fundagio Calouste Gul-benkian (Portugal), Fundagio Luso-Americana para o Desenvolvimento (Portugal), and the U.S. Department of Energy (D.O.E.) under cooperative research agreement

Contents

1 Introduction

2 Super-Yang-Mills Theory

2.1 Introduction . . . . 2.2 Review and Conventions . . . . 2.3 Canonical Formulation and GL(3) Properties . . . . 2.4 Geometric Variables . . . . 2.5 Gauge Tensors as Geometric Tensors . . . . 2.6 Conclusions . . . .

3 Bosonic a--Models

3.1 Introduction . . . .

3.2

Order a' . . . .

.

3.3

O rder a'

2

. . . .

4 Heterotic c--Models

4.1 Introduction . . . .. . . . .

4.2 Duality in the Heterotic Sigma Model . . . . 4.3 Renormalization and Consistency Conditions . . 4.4 Duality, the Gauge Beta Function and Heterotic 4.5 Torsionfull Backgrounds . . . .

4.6 Conclusions . . . .

Anomalies

5 The Gravitational Axial Anomaly

7

11

11 15 18 24

30

38

41

41

43

45

51

51 54

57

60

64

66

69

5.1 Introduction and Discussion . . . . 69

5.2 The Abelian Higgs Model and Conformal Symmetry . . . . 72

5.3 The Three Point Function for the Two Loop Gravitational Axial Anomaly 78 5.3.1 Diagrams in Figures 1(b) and 1(c) . . . . 79

5.3.2 Diagrams in Figures 1(d), 1(e) and 1(o) . . . . 84

5.3.3 Diagrams in Figures 1(f), 1(g) and 1(h) . . . . 86

5.3.4 Diagrams in Figures 2(a) and 2(b) . . . . 88

5.3.5

Diagrams in Figures 1(i), 1(j), 1(k), 1(p), 2(c) and 2(d)

. . .

91

5.3.6 Diagrams in Figures 1(1), 1(m) and 1(n) . . . . 93

5.3.7 The Three Point Function . . . . 94

A Kaluza-Klein Tensor Decompositions 99

B Differential Regularization 101

Chapter 1

Introduction

Ever since the dawn of modern science, the fields of physics and mathematics have been unequivocally associated to each other in a multitude of situations and areas. One may possibly claim that they also share the same roots, and therefore advances in one field must always reflect on the other and vice-versa, no matter how trivial or fundamental such a reflection might take shape. Examples of such situations are quite often met in the research which is nowadays performed (independently) in both

fields.

Having followed on somewhat distinct paths perhaps ever since late in the last century, there is still a very strong interest by many researchers in the boundary of the two fields, exploring the interface science that has come to be known as mathematical physics. One such aspect that we wish to explore in this thesis is what lies in this interface at the point where Riemannian geometry and quantum field theory meet. We shall see, through the three distinct problems that build this thesis, that many interesting results are there to be explored and investigated.

We shall begin by looking at a problem in 3 + 1 dimensional supersymmetric gauge theory, to be specific, A = 1 supersymmetric SU(2) Yang-Mills theory. In here we develop a new tool to study the strong coupling limit of this theory, in the form of introducing new variables for the Yang-Mills theory, which have the property of being gauge invariant. Indeed gauge invariance is an important constraint on the states of the gauge theory, in the form of Gauss' law. The fact that the Yang-Mills

system is constrained has been a difficult drawback to solve in order to fully explore the quantization of this theory. The question of whether one can construct gauge invariant variables then becomes of relevance as one realizes that such variables would allow for a trivial implementation of the Gauss' law constraint. If moreover one can construct these variables such that they are local, they would then seem to be the most appropriate ones to describe the moduli space of the theory. One more point in favor of such a programme is the fact that in temporal gauge the remaining gauge invariance of the Yang-Mills theory is restricted to space-dependent transformations at a fixed time. This is in fact the true quantum mechanical symmetry of the theory. Working with local gauge invariant variables this symmetry of the Hamiltonian can be maintained exactly, even under approximations to the dynamics.

All this said, we strongly believe that this is indeed an interesting problem to ex-plore in quantum field theory, but one would not seem to realize where the connection to Riemannian geometry would come into the game. What we shall see later is that such a connection arrives from the way we will choose to define the new variables: we shall replace the gauge connection of SU(2) by a covariant variable under the gauge group, which shall enjoy the fact that it can be also interpreted as a dreibein, i.e., a square root of a metric. We shall see that this metric lives in a 3 dimensional mani-fold, and that it can be used as a local gauge invariant variable for Yang-Mills theory. However, our interest in here is, as we mentioned before, on supersymmetric Yang-Mills theory. Therefore, we must not forget to include the fermionic partners of our bosonic variables. In chapter 2 we shall see in detail how this can be accomplished. We shall learn that the local gauge invariant variables we will construct for V = 1 supersymmetric Yang-Mills theory have a Riemannian geometrical interpretation in the sense that they can be constructed such that the emergent geometry is that of

K

1 supergravity: a Riemannian geometry with vector-spinor generated torsion. After studying this problem, we leave gauge theory behind and move into the do-main of sigma models, where we shall study both bosonic and supersymmetric sigma models. In these models, describing maps from a given two dimensional surface into a general target manifold, Riemannian geometry makes its appearance from the very

beginning, in the action for the models we shall consider. Indeed, having emerged from string theory - a possibly quantum theory of gravity - it is but to expect that a metric should somehow be incorporated in these models from scratch. Indeed, Rie-mannian metrics in the target manifold are nothing but infinite-dimensional coupling parameters of the two dimensional quantum field theory. The Riemannian geometry of the target is therefore constrained by the quantum field theoretic properties of the two dimensional theory and, in the particular case of string theory, the condition that the beta functions for the diverse couplings of the sigma model vanish is equivalent to saying that the geometrical structures in the target manifold obey the

[generalized]

Einstein's equations.

But again because these models are string theory inspired, we can look at all the nice properties of strings and ask which, if any, of such properties are still valid once we move away from the conformal fixed points where the sigma models describe strings - and in particular whether such properties have any chance of being valid throughout all of the parameter space of the sigma model. One such property we shall be interested about, and which we shall study in detail in chapters 3 and 4 of this thesis, is target space duality, henceforth T-duality. This is a perturbative symmetry of string theory which basically relates target manifold compactifications in circles of radius R with compactifications in circles of radius trg ,/R, with trg being the characteristic string length. We will learn that by exploring a consistency requirement between T-duality and the renormalization group flows of the sigma model, we shall be able to find the beta functions of these models for all the coupling parameters. From a string theory point of view this simply means that geometrical target space properties are encoded in this T-duality symmetry. Moreover, in the case of heterotic sigma models, we will also learn that this duality symmetry provides new insight and mechanisms for cancelation of a certain class of anomalies.

Once we are done with sigma models, we shall return to the realm of the 3+1 world, and study the problem of gravitational axial anomalies. If the anomaly is beyond doubt a quantum field theoretic phenomena, it is also not less clear that gravitation is a Riemannian geometrical phenomena (at least in the domain of energy we shall

be looking at). We are therefore in a situation where we encounter quantum fields in a curved background spacetime. In this last chapter of the present thesis, chapter 5, we shall compute a radiative contribution to an anomalous correlation function involving one axial current and two energy-momentum tensors, corresponding to a contribution to the gravitational axial anomaly in the massless Abelian Higgs model. We shall learn new techniques to perform such a complicated calculation, and we shall see that the two loop contribution is found not to vanish, due to the presence of two independent tensor structures in the anomalous correlator.

The three problems dealt with in this thesis are clearly quite distinct, but they all share the property of presenting yet some new examples of interactions between Riemannian geometry and quantum field theory. These problems appeared in the literature as four distinct publications. Chapter 2 was published in Nuclear Physics, [65]. Chapter 3 was published in Physical Review Letters, [47], and chapter 4 is to be published in International Journal of Modern Physics, [58]. Finally, chapter 5 was published in Physical Review, [60]. During the process of five years of study at MIT, I also enjoyed the opportunity of doing other research, not directly related to this thesis. In particular, other matters and problems were studied, and I believe they should be mentioned in here. These research projects were not included in this thesis as they did not share the same theme studied in here, the one of interactions between geometry and quantum theories. These projects were (a) studies on classical configu-rations of string theory in 3+1 dimensional target manifolds, where the strings under consideration had an initial knotted topology. These investigations were published in Physics Letters, [24, 66]. The other research project was (b) a study of the quantum cosmology of an S-duality invariant V = 1 supergravity model in a closed homoge-neous and isotropic Friedmann-Robertson-Walker spacetime, and which is to appear in Classical and Quantum Gravity, [16].

Chapter 2

Super-Yang-Mills Theory

2.1

Introduction

For quite sometime now there exists a nice geometrical setting for Yang-Mills theory. That is based on fiber bundle differential geometry, where the configuration space is obtained by factoring out time independent gauge transformations, and is then seen as the base space of a principal fiber bundle, where the structure group is the gauge group

[57].

There are many concepts of Riemannian geometry that can then came into the game, as there is the possibility of defining a Riemannian metric on the space of non-equivalent gauge connections

[8].

However, this setting must be cast into a more workable form when we want to study the strong coupling regime of Yang-Mills theory. In here, gauge invariance becomes an important constraint on states of the theory in the form of Gauss' law. This constraint amounts to a reduction of the number of degrees of freedom present in the gauge connection: if one starts with a gauge group G, in the canonical formalism and in temporal gauge A' = 0, the number of variables is 3 dim G, when in fact we only have 2 dim G physical gauge invariant degrees of freedom. The question of whether one can construct local gauge invariant variables is then an important one, as it would allow us to easily implement the Gauss' law constraint. These variables would then seem the most appropriate ones to describe the physical space of the theory. Moreover, observe that in temporal gauge the remaining local gauge invariance is

now restricted to space-dependent transformations at a fixed time. This is the true quantum mechanical symmetry of the theory. Working with local gauge invariant variables, this symmetry of the Hamiltonian can be maintained exactly, even under approximations to the dynamics.

This idea first appeared in [48, 36], and has recently gained new momentum with the work in [30, 11, 54, 43, 55, 40, 33, 45, 44], and references therein. In [43], one constructs a change of variables that will allow replacing the coordinates A by new coordinates u7 which have the property of transforming covariantly under the gauge group, as opposed to as a gauge connection. Then, in these new coordinates, the generator of gauge transformations becomes a (color) rotation generator, and by contracting in color we can obtain gauge invariant variables to our theory, gij =

uqu. _{States I[gi-] depending only on these gauge invariant variables manifestly}

satisfy Gauss' law. One must be careful, however. Not any choice of gauge covariant variables is adequate: an appropriate set of variables should describe the correct number of gauge invariant degrees of freedom at each point of space, and should also be free of ambiguities such as Wu-Yang ambiguities [79]. In this case, several gauge unrelated vector potentials may lead to the same color magnetic field. Variables that are Yang insensitive are of no use, as in the functional integral formulation Wu-Yang related potentials must be integrated over - _{since they are not gauge related -,}

while functional integration over Wu-Yang insensitive variables always misses these configurations. The absence of Wu-Yang ambiguities will be clear if we are able to invert the variable transformation, i.e., if when transforming A --+ u one can also have an explicit expression for A[u].

In [43], the set of gauge covariant variables

{ui}

that replace the SU(2) gauge connection was defined by the differential equations:

which is equivalent to writing,

--

+

6abcA bUc -

Is

ak =

0,

(2.1.2)

as the

{ua}

have det u : 0, and so form a complete basis. Observe that we have fje - ig, and these quantities can be written as,

1

3jk g"(&g gn + kgkYa -

angk),

(2.1.3) where

g ui= tut. (2.1.4)

So, a "metric" tensor was implicitly introduced by the defining equations for the new variables, (2.1.1). Observe that equation (2.1.2) is simply the so-called dreibein postulate, where the

{u}

plays the role of a dreibein, Lif = C₆ A. is a spin-connection, and I . is the affine metric connection. A torsion-free Riemannian

geometry in a three manifold was then introduced by the definition of the new vari-ables. The metric gij contains in itself the six local gauge invariant degrees of freedom of the SU(2) gauge theory. Moreover, any gauge invariant wave-functional of Ai can be written as a function of gij only, and any wave-functional of gi, is gauge invariant [43]. This implements gauge invariance exactly. Finally, the dreibein postulate can be inverted so that one obtains,

1

bc=

2 bi

*V UC

(2.1.5)

where we use the notation V u - n U for the purely geometric covariant derivative (as opposed to the gauge covariant derivative). Therefore, the new variables avoid Wu-Yang ambiguities.

Full geometrization of Yang-Mills theory in this formulation was then carried out in [43, 40]. The electric energy involves the inversion of a differential operator that can generically have zero modes. By deforming equations (2.1.1) it was then shown how one could proceed to compute the electric tensor

[45].

Instanton and monopole

configurations have been identified as the S3 _{and S}2 _{x R geometries [45], and, more} recently, the form that the wave-functional for two heavy color sources should take has been calculated [44]. The computations are carried out in the Schr6dinger repre-sentation of gauge theory, see [51] for a review.

Supersymmetric Yang-Mills theory has also been well established for quite some-time now. It allows for many simplifications in quantum computations, and with an appropriate choice of matter content and/or number of supersymmetry generators, one can obtain finite quantum field theories. Textbook references are [9, 77, 78].

Moreover, recently there has been a lot of progress and activity in the field due to the possibility of actually solving for the low-energy effective action of certain cases of supersymmetric Yang-Mills theories, starting with the work in [68]. It is then natural to extend the work on gauge invariant geometrical variables to the supersymmetric case. That is what we shall do in here.

We shall see that it is possible to define variables that also have a geometrical interpretation, namely, as the variables present in supergravity. We should point out, however, that no coupling to gravity is ever considered. Still, we need a motivation to construct the new variables. As in the pure Yang-Mills case the new variables and geometry have an interpretation as the variables and geometry of three dimensional gravity, it is natural to assume that in the supersymmetric case the new variables and geometry could likewise have an interpretation as the variables and geometry of three dimensional supergravity.

This shall be a guiding principle throughout our work. More geometrical intuition on how to construct the new variables will come from an extra symmetry enjoyed by both the canonical variables and Gauss' law generator. That is a symmetry under

GL(3) transformations, a diffeomorphism symmetry. This will allow us to naturally

assign tensorial properties to diverse local quantities of the theory. Obviously the Hamiltonian (or any other global operator) will not possess this symmetry. After all, supersymmetric Yang-Mills theory is not diffeomorphism invariant.

The plan of this chapter is as follows. In section 2 we start by reviewing the conventions of V = 1 supersymmetry, and also outline the geometry of supergravity.

In section 3 we will then explore the GL(3) symmetry, assigning tensorial properties to local (composite) operators. With this in hand, we then proceed to define gauge invariant geometrical variables for supersymmetric Yang-Mills theory in section 4, carrying out the full geometrization of the theory in section 5. Section 6 presents a concluding outline.

2.2

Review and Conventions

The conventions in [9, 77, 78] are basically the same. We will follow [9] with minor changes, as we take a"' = 1[/,"]. The

K

1 supersymmetry algebra is obtained by introducing one spinor generator, Q, which is a Majorana spinor, to supplement the usual (bosonic) generators of the Poincard group. The = 1 supersymmetry algebra is then the Poincard algebra plus:

[P

1

, Q] =

0,

[I,,, Q]

=

-iaVQ,

{Q, Q}

=

2-y"P,,

(2.2.1)

where Q --

Q

0.

Supersymmetric gauge theory, based on gauge group G with gauge algebra

g,

has as component fields the gluons, or gauge connection, A a; the gluinos, super-partners of the gauge fields and Majorana spinors, Aa; and the scalar auxiliary fields Da. All these fields are in the adjoint representation of

9.

In Wess-Zumino gauge, the action is,

1i

Sa= dF {--F,

"+

"DA +

D

(2.2.2)

4

~

22

where we can see that the auxiliary fields have no dynamics. The supersymmetry transformation laws of the fields, that leave the action invariant are:

6A"a = (o-"'F" -ZiysDD)a

6D

a = E-/5-/DPA a,

_(2.2.3)

where E is a Majorana spinor, which is the parameter of the infinitesimal supersym-metry transformation. These transformation laws implement a representation of the

.I= 1 supersymmetry algebra in the quantum gauge field theory. The Noether conserved current of supersymmetry is a vector-spinor,

JP = iy o ,0F,")a (2.2.4)

and so the quantum field theoretic representation of the supersymmetry generator is given by the Majorana spinor,

Q

= ild 3X 70-"'F,, Aa. (2.2.5)

This outlines our usage of notation for supersymmetric gauge theory. We still have to outline notation for the supergravity geometry. In here, one has a graviton,

g,,, and a gravitino which is described by a Rarita-Schwinger field, 0, So, we need

to start by reviewing notation for inserting spinors in curved manifolds. Having a metric, one can define orthonormal frames and so insert a tetrad base at the tangent space to a given point, which will allow one to translate between curved and flat indices. In particular, this allows us to introduce gamma matrices in the manifold, and so introduce spinors. If we consider a manifold M, and pick a point p E M, we can introduce a tetrad base {tUt} at p via,

Yiv _UYvab,ab _(2.2.6)

defining an orthonormal frame at each point on M. One can now insert gamma matrices as )4(x)ua(X) = ya, where the ya are numerical matrices. Local Lorentz transformations in the tangent space T M are Aab(p) and (Dirac) spinors at p E M

rotate as,

Oc(p) - SoO(A'b(P))OO(P). (2.2.7)

Next, one constructs a covariant derivative, Da4', which is a local Lorentz vector, and transforms as a spinor,

Da4a -* Sac(A)AabDb4)o. (2.2.8)

That is done via a connection Q. such that, Da' = u +(ag

+

Q,)4, and,

1 1 b

W = gbab = a "v PU b Uab,

(2.2.9)

where wab is the spin-connection.

Now that we have spinors defined on curved manifolds, we can proceed with supergravity. In here, the Riemannian connection FP, is not torsion-free. It is still metric compatible, so that one can write,

:PP = FP - K , (2.2.10)

where Fr is the affine metric connection, and K, ' is the contorsion tensor. Hatted symbols will always stand for quantities computed via the affine metric connection. The torsion tensor is,

TZ V = FP - IF, (2.2.11)

and so,

Kv = (T,₁

V

- YvA gT T'a A- gAg"pTv A). (2.2.12)

2

In

K

= 1 supergravity, the torsion is defined by the Rarita-Schwinger field 0, as,

Ta

V

= k O>_f o" (2.2.13)

2

postulate is,

'

-

(9U, ua + ab - p a

-0

(2.2.14)

and the covariant derivative acting on spinor indices is,

(D4)ao _

6o4,0

+ IWiab(0

ab)c.

(2.2.15)

Finally, the supersymmetry transformations that leave the

KV

1 supergravity action (Einstein-Hilbert plus Rarita-Schwinger) invariant are:

&u",(x) =ifx $()

60P() = 2D(x),

(2.2.16)

where _{(x) is the infinitesimal parameter of the transformation (now a space-time} dependent Majorana spinor), and where we have not included the auxiliary fields. This ends our review and outline of conventions. We can now start analyzing the gauge invariant variables geometrization of supersymmetric Yang-Mills theory.

2.3

Canonical Formulation and GL(3) Properties

In the Lagrangian formulation of the theory, the

K

= 1 supersymmetry algebra closes

only up to the field equations. In order to obtain manifest supersymmetry, and off-shell closure of the algebra, one needs to introduce auxiliary fields. In contrast to this situation, it is known that in the canonical formalism the

KV

1 super Lie algebra closes without the introduction of auxiliary fields (in terms of Dirac brackets the algebra closes strongly; otherwise it closes weakly, i.e., up to the first-class constraints)

[73, 72]. So, we drop the auxiliary fields.

The Hamiltonian for supersymmetric gauge theory is therefore,

where e _{is the coupling constant. The gauge covariant derivative is,}

D-Aa = &jAa + fabcA Ac, (2.3.2)

and the magnetic field potential energy,

Bi [b]2 Ik F," A ] = b iJk (A + fabcAbAc). (2.3.3)

We still have to impose the Gauss' law constraint on the physical states of the theory,

ga(x)

D-Eaz(X)

-

facA (x)Ac(

g((x)[A.,

Ac]

0.

(2.3.4)

2

This local composite operator is the generator of local gauge transformations.

There is one more element in the

K

1 supersymmetric Yang-Mills theory, and that is the Majorana spinor

Q,

the generator of supersymmetry. Using the definition,

Q

J

d3X

Q(x), (2.3.5)

one can then write,

Q(x) = i(-ey;E i(X) + -Eijk100u ZB"k[Ab(x)])A"(x), (2.3.6)

or, using the explicit Weyl representation of the gamma matrices, we can equivalently write this local composite operator in a more compact form,

0

(eEai(x)

+

'Bai[Ab(x)])o-)

Q W

(-eE

i(X) +

B

at

[An

(X)])g- 0 ()

(2.3.7)

In the bosonic half of the theory, the canonical variables are A?(x) and Ea,(x). Canonical quantization is carried out by the commutator,

The momentum E"i(x) will be implemented as a functional derivative acting on wave-functionals as,

Eai(X)4[A',

Ac]

--

+

-i

x[A',

Ac].

(2.3.9)

n

6A

(X) n

In the fermionic half of the theory, one has a Majorana spinor Aa(x). Canonical quantization is carried out by establishing the anti-commutation relations for the spinorial field,

{A'(x), A'(y)}

= 6

( - y). (2.3.10)

Both the commutator and the anti-commutator are to be evaluated at equal times. We can now compute the commutators and anti-commutators of this theory, which involve the composite operators H, ga(x) and Q(x). Clearly, these (anti)-commutators are

related to the symmetry transformations generated by these operators.

The commutators involving the generator of local gauge transformations of the canonical variables can be computed to be,

[A(x), Gby)

-

i(&ba

-

f"cbAc(x))6(x

-

y),

(2.3.11)

[Eai(X), g b(y)] = f abcEci(x)6(x - y), (2.3.12)

[Aa(x),

gb(y)]

=

ifab"A(x)6(x

-

y),

(2.3.13)

and the (anti)-commutators involving the local composite operator associated to the supersymmetry generator can similarly be found to be,

[A'(x), Q(y)] = -- ;Aa(X)6(X - y), (2.3.14)

[Eai(x), Q(y)] = e(6knm 0 nm)DjAa6(X -

y),

(2.3.15)

1

{Aa(x), Q(y)} = (-eyE i(X) + -gk ujB ak[A (X)])6(X - y). (2.3.16)

gener-ator are both gauge invariant composite opergener-ators, as,

[H, ga(x)] = 0, (2.3.17)

[Q(x), g(y)]

=

0.

(2.3.18)

As expected, the generators ga(x) define the local gauge algebra,

[ga(x), g()] = fabcgc(x)6(x - y)

(2.3.19)

The supersymmetry generator

Q

defines, along with the generators of the Poincar6 algebra, the g = 1 supersymmetry algebra [73]. However, the defined local composite operator Q(x) does not define a local algebra. That is to be expected as we do not have local supersymmetry in the theory. This local operator was only introduced in order to facilitate the following tensorial analysis based on diffeomorphism transformations of the presented (anti)-commutators.

So, we now want to check that there is a GL(3) symmetry at work for the formulae (2.3.8), (2.3.10) and (2.3.11-13), (2.3.19). The bosonic part tensorial assignments will be just like in the pure Yang-Mills case [43], as is to be expected. The mentioned canonical relations are covariant under diffeomorphisms xz - yn(X,) on the domain

R3_{, provided Ai(x) is a one-form in R}3_{, transforming as}

axi

A'"(y') = Aa(), (2.3.20)

where [axi/9yn] is a GL(3) matrix. That A(x) = Aa(x)Tadx" is a Lie algebra valued one-form is a well known fact from the fiber bundle geometry of gauge theory; so consistency holds. Also, provided Eai(x) is a vector density (weight -1) in R3_,

transforming as

E'an(y"n) =

det[

]--Ei(x).

(2.3.21)

The same property holds for Bak(X). This is consistent with the implementation of Eai(x) as a functional derivative (2.3.9), and with the definition of the magnetic

field (2.3.3). Commutator (2.3.8) is then clearly diffeomorphic invariant, without the intervention of a space metric. However, to introduce spinors, one does need a metric (more precisely, a dreibein base). We shall assume there is a metric, gij, and later we will construct it using the bosonic dynamical variables of the theory.

When restricted to three dimensional Euclidean space, Lorentz transformations become rotations in R3_{. The spinor representation of a rotation is then, at a point}

p E M, given by the orthogonal matrix acting on spinor indices,

1

Sap(A(p))

=

exp( wab(A(p))Uab)0,,

(2.3.22)

2

where w is the rotation parameter. We can now define the GL(3) properties of Aa(X), in order to maintain the anti-commutator (2.3.10) diffeomorphism invariant. That relation is invariant under diffeomorphisms, provided Aa(X) is a spinorial density (weight - ) in R3, transforming as

axi

A'"(ym) = det[O ]7Soy (A(p))AO(xi).

(2.3.23)

Let us see what are the consequences of these GL(3) properties on the composite local operators ga(x) _{and Q(x). Starting with the generator of local gauge}

trans-formations, we observe that the tensorial properties of the canonical variables imply that under diffeomorphisms one will have that ga(X) _{is a scalar density (weight -1)}

in R3, transforming as

I

OX

_(2.3.24)

" )

det[]g ().(

This automatically verifies that the canonical commutators (2.3.11-13) and (2.3.19) are invariant under local diffeomorphisms on the domain of the local canonical vari-ables.

Now, look at the other local composite operator, Q(x), (2.3.6) or (2.3.7). First observe that in (2.3.7) the Pauli matrices a are numerical matrices, and not dynamical ones (in which case one would write o()u?(x) =-,

{u}

being a dreibein base).

Write (2.3.7) as:

Q(x) - oa

W

L i(x),

(2.3.25)

where,

0

eEai(x) +

'Bai[An(x)]

U, (X)'X)e + az*

A

a(X).

(2.3.26)

-eEgix +B B[A'(x)]

0

Then, the tensorial properties under GL(3) of the canonical variables imply that, under diffeomorphisms, one will have that H(x) is a vector-spinor density (weight -) Din R3, transforming as

U'e (y" ) = det [

]i -ScO(A (p)) I'(x')

(2.3.27)

However, as the ai's in (2.3.25) are numerical, they do not transform under the diffeomorphism, and so Q(x) fails to be covariant. This is to be expected, as we will see below.

The GL(3) symmetry of the (anti)-commutation relations involving local (com-posite) operators and local variables has been established, given the tensorial prop-erties assigned to the canonical variables. Clearly, the theory itself fails to be GL(3) invariant, and that is to be expected: the Hamiltonian is not covariant under diffeo-morphisms

(the

metric

6,j

appears instead of gi, the measure d3x appears instead

of "Fgd3X, etc.). This can be related to the lack of covariance of the supersymmetry

generator (2.3.25), (2.3.27). Indeed, one can regard Q(x) as the square root of the Hamiltonian; and so if the Hamiltonian fails to be covariant, so should the supersym-metry generator. Moreover, observe that when we square (2.3.25) we will obtain a term like uicb = 6ij + Zejkck, and this can be seen as the origin of the "wrong" metric

61, in the Hamiltonian, which shall destroy the possibility of local covariance. Also,

no global (composite) operator can have this GL(3) symmetry, due to the "wrong" choice of integration measure. Now that we assigned tensorial properties to local quantities in the supersymmetric theory, we are ready to proceed in looking for fur-ther geometrization in this canonical framework.

2.4

Geometric Variables

We shall limit ourselves to the simplest case of non-Abelian gauge group, namely,

G = SU(2). Then, the structure constants are simply fabc - Cabc. We will assume knowledge of the previous work done for pure Yang-Mills theory [43, 40, 45].

One wants to have a representation of supersymmetry once we are to transform to the new variables. As we know from the bosonic case [43], the gluon field is transformed into a "metric" field. The supersymmetry representation that includes a metric field is that of supergravity, and it also includes a vector-spinor field. So, one will expect that the gluino field will be transformed into a "gravitino" field. We shall therefore wish to transform the supersymmetric Yang-Mills variables,

_{A?(x),

Aa(x)},

into the variables of three dimensional supergravity,

{yiJ(X),

4k(x)}. We shall also expect to obtain a geometry similar to the one of supergravity. After all, the defining equation for the

{ua(x)}

variables identifies them with a dreibein base in a three dimensional manifold.

Recall form section 2 what one is to expect. The geometry will have torsion, defined as,

T =0 I ."$a3. (2.4.1)

We can insert a dreibein base through,

gi3 U'ab~ab (2.4.2)

and also expect that there could be some local supersymmetry transformation in these new variables, which we shall call the "geometrical supersymmetry variation", and which would look like the supersymmetric transformation laws of supergravity,

(u"(x)

= if(X)-";x),

64k(x) = 2Dk

(x),

(2.4.3)

this at hand, the dreibein postulate is now written as,

D at" = + Wabub - an = 0, (2.4.4)

also defining the operator Dj. Multiplying this equation by cik, and defining the spin-connection via the gauge connection, as in,

Wab(X ab cA'(x),

(2.4.5)

the dreibein postulate becomes,

ij k = ₆

ik

a+ bcAb, k) -

2j

Tika=4C k

(2.4.6)

We shall take these differential equations to define the change of variables A

(x)

-ug(x). Then, the reverse line of argument holds: the new variables

{ua(x)}

play the role of a dreibein, and from them one can construct a metric gi =y uau which is a local gauge invariant variable. The geometry defined by this new variable has torsion, given by (2.4.1). Clearly, for the change of variables to be well defined, we still need to specify what O4(x) is. That is the problem we shall now address.

Let us begin with some dimensional analysis. We know that the gauge field Aa(X) has mass dimension one, and the gaugino field Aa(x) has mass dimension three halfs. We also know that the mass dimension of the fermionic generator of the supersym-metry algebra is one half. Through definition (2.3.5) and expression (2.3.25) one observes that if one is ever to modify the gauge theory in order to covariantize it (in-serting

o(x)u?(X)

= a in

(2.3.25)

and from then on), one would require the dreibein field to have zero mass dimension, as well as the metric. Though we are not going to modify the gauge theory in this work, we may as well stick to this broader per-spective. Then, through the dreibein defining equation (2.4.6), we conclude that the "gravitino" field has mass dimension one half. These dimensional assignments are just like what happens in supergravity.

"gravitino" defining equation. In fact, there are some a priori requirements for such an equation. It must be geometrical, either in a differential or algebraic way; one needs 12 equations, to change the 12 variables A a to the 12 variables bk,; and the gluino field must be present in such an equation. If we moreover require linearity on fermionic variables (like we had linearity on the bosonic variables in (2.4.6)), we see that, by simple dimensional analysis, we can not write such an equation algebraically, but only differentially. Moreover, the equation is constrained to be of the form,

3 = M CD'Oko a

A

, (2.4.7)

where the matrix Aia must have zero mass dimension, being so far otherwise arbi-trary. However, one must be cautious. Not only do we want to have a geometrical way in which to define the vector-spinor field, but we also want to be compatible with the fact that we are studying a supersymmetric theory. In particular, we would like the geometrical supersymmetry variation (2.4.3) to generate the gauge supersymmetry variation (2.2.3). So we shall ask for the geometrical variation (2.4.3) to generate the gauge supersymmetry variation on the bosonic variables A?(x), and in the simplest case where (x) = E.

Under a generic variation of the fields, one obtains for (2.4.6),

ei3'kDi = -ci3keacuc bcU" Ab+ 2i 3 ' , (2.4.8) where,

a - ij 3_,a 60k. (2.4.9)

2 2

The supersymmetry transformation laws we shall need are (2.2.3) and (2.4.3). So, a supersymmetry transformation of the dreibein defining equation yields the "gravitino" defining equation. Performing the computations, based on the previous formulae, we are led to,

where ' i(x) = u (x)ya.

We shall take these differential equations to define the change of variables Aa(x) + /k(X)- Observe that this equation is precisely of the required form (2.4.7), and the matrix Mia has been uniquely defined. Also, this alone guarantees that the geometric variation (2.4.3) will generate the bosonic gauge supersymmetry variation (2.2.3), when - e. This does not guarantee however that the geometric variation will generate the fermionic gauge supersymmetry variation under the same circumstances. In fact, we can choose -

[E]

through a differential equation (2.4.20) for , such that the geometric variation generates the gauge supersymmetry variation on Aa(x), but we shall not have - E in this case. This shows that, even though we can generate the

gauge supersymmetry variation via the geometrical supersymmetry variation under special circumstances, the geometric variation is not the original supersymmetry of Yang-Mills theory. The actual expressions for the supersymmetry variations on the new geometrical variables can nevertheless be computed using the usual expression,

&D =_ i[Q,

ID],

(2.4.11)

where <D is any of the geometrical variables, and where we should express the su-persymmetry generator in this geometric framework (see section 5). The resulting expressions would not be as simple as (2.2.3) or (2.4.3).

All together, one sees that we can now define local gauge invariant geometric variables for supersymmetric Yang-Mills theory via the system of coupled non-linear partial differential equations, (2.4.6) and (2.4.10). These equations define a variable change

{A?,

Ab} --+ {Ua, 4

k}. They also introduce a three dimensional Riemannian

geometry with torsion as given by (2.4.1-2) and (2.4.4).

Now that the definition of the new geometrical gauge invariant variables is con-cluded, one would like to invert the defining equations, in order to express Aq(x) and Aa(x) in terms of the geometric variables. This inversion will make it clear that there are no Wu-Yang ambiguities related to these new variables. The defining equa-tion for the dreibein (2.4.6) is equivalent to the dreibein postulate (2.4.4), where the

connection is with torsion,

J'k = ' -K Ak, (2.4.12)

hatted symbols always denoting affine metric connection quantities. The contorsion tensor is computed from the torsion tensor, through (2.2.12), and one obtains,

Kijn,= (Vyj On + j y7bn - n ) (2.4.13)

Define a purely geometric derivative through,

V7ju" -,u'

-

F'u',

(2.4.14)

and we can find the expression for the inversion,

A

(x) = abc Ubk (X)V c (X).

2k

(2.4.15)

We shall next compute a generic variation of this equation, so that one can later use it to compute the inversion for the gluino field. In order to carry out the calculation, we will need to know what is the generic variation of the connection (2.4.12). Using the fact that it is metric compatible, this can be computed to be,

1 m

6n = I g" (V &gmk + Vkggmj -- VmSg'k) - AKjgk. (2.4.16) One can now carry out the variation of the dreibein postulate,

the variation of equation (2.4.15). The result is,

A.=

Umin _z a(V7(Ubn

_M

_z2

_2,

_,j

+ Vi(1&gni) + >((i/ ₁ )&un +

+2(bn77i/$ + '&m8b + ibyi6ii)),

and from there obtain

1

2 (n 0)U)

(2.4.17) where lg = det u.

variation (2.4.3) with = E generates the bosonic supersymmetry variation. So, we

only need to use (2.2.3) and (2.4.3) in (2.4.17), and rearrange, so that we find the expression for the inversion,

Cnml

Aa(x) = - U,-(X)

Y(X)

(7(X)ViOn(X)+7n (X)DIZ(X)+-i(X)DIn(X)),

(2.4.18)

6 g(x)

where one uses the vector-spinor full covariant derivative, defined as, 1

Di pka - Di?/ka + 1W b(ab)c ?kO - liksa. (2.4.19) 2

Observe that even though the spin-connection is defined via the gauge connection, it is a fully geometric quantity through the dreibein postulate. Later on we shall also require an expression for the generic variation of this equation, so we will address such a problem now. The computation is rather long, and so is the result. One obtains,

6nml

- u(ubi 6U + U,6ubi 7b "(Din - Du

pi)-6 _g

'nml

- _{u~f('(iD(6b) +} M^/Di(Si) +

Ti(i(n)-1

2 k nVi +Ynik ' O V

+

ik $nVi)(u bk

6z)-1 J

-- i 'knV(6gki) + TN Orko4Vj(6gkl)

+

7yjik4nVj(6gkI))+

2 1

+ (i-I k In6Kijk + 7ng ' ikp)i&Alk +71_Cjk On6Ki'k) + (7Kins + -Yn6Kii" + Kn)s)-2

1

- u~ui(cnik ucl U CD - clikoucn j( iDi<$n

+

'n~h4i

+

in ), (2.4.20)

where the generic variation of the contorsion tensor can be written as,

6Kmin= -(($,7a4i)6ua +

(4

a4p,)6qU - (4ya4,a) 6ua)+

+-((0n-/ -

i-/)601

+ (OiN + 4n-yi)6S1 - (01'i' +

4,'yi)60n).

(2.4.21) 4

The variations (2.4.17) and (2.4.20) allow us to express a variation of the wave-functional in terms of the variations of the geometric variables. This will be helpful in section 5.

The inversion completed proves the non existence of Wu-Yang ambiguities in the new geometrical variables. Therefore, we have managed to define new gauge invariant variables for supersymmetric Yang-Mills. Moreover, it can be shown that gauge invariant physical wave-functionals of the theory depend only on these geometric variables

(see

section 5),

1V

'I[giJ, Ob], so that we have in these variables an explicit

parameterization of the physical Hilbert space (moduli space) of the gauge theory. A final remark on diffeomorphisms is now in order. As said before, only the variables of the theory are diffeomorphism covariant. The Hamiltonian fails to be diffeomorphism covariant. Given that the variables of the theory are now

{gi,

O

},

this has an

interesting consequence: a configuration diffeomorphic to the previous one yields a different configuration to the gauge theory. Therefore, we can extend solutions to the gauge theory by action of the group of diffeomorphisms, by simply moving along the orbit of the geometrical configuration.

2.5

Gauge Tensors as Geometric Tensors

We now wish to write the tensors and composite operators of our theory in terms of the new geometric variables, i.e., as geometric tensors and geometric composite operators. We shall first address the electric and magnetic tensors. The Hamiltonian, Gauss' law generator, and the supersymmetry generator composite operators then easily follow from these two tensors and the previous equations for the inversions of the gluon and gluino fields.

Let us start with the gauge Ricci identity,

and apply it to the dreibein field. We will obtain,

abcjF bu =

R k ua,

(2.5.2)

where Rki3 is the Riemann tensor of the connection F,

-~i 0J_

~

+i' J7F- - F~Fl (2.5.3)

From (2.5.2) one can express the field strength in terms of the Riemann curvature, and so from (2.3.3) we can express the magnetic field vector geometrically, as,

B - 1 miJC nk UaRlk' .

(2.5.4)

So, the gauge invariant tensor which gives the Yang-Mills magnetic energy density is,

Ba'Ba

=

1

imn CiklRuvmn(Ruvkl -

Rvukl).

(2.5.5)

16

As we can see, this expression gives the gauge invariant tensor in a manifestly gauge invariant form, in terms of the "metric" gji, and the "gravitino" 4

k (which is present

via the torsion contribution to the Riemann tensor).

The electric field vector is the momentum canonically conjugated to the canoni-cal variable, the gauge connection. In canonicanoni-cal quantization it is represented by a functional derivative (2.3.9). We define a gauge invariant tensor operator eij by,

6A?(x)

=

iEai(x) =

g(x)ut()eiJ(x).

(2.5.6)

Clearly, eiJ(x) is an ordinary

(')

tensor under GL(3). From this expression, the electric gauge invariant Yang-Mills tensor, i.e., the manifestly gauge invariant tensor which gives the Yang-Mills electric energy density, now follows as,

In order to finally obtain the Hamiltonian in a manifestly gauge invariant form in terms of the geometrical variables, one still needs the expression for the fermionic energy density, as is clear from (2.3.1). The expression for this gauge invariant tensor can be obtained by simply inserting (2.4.18-19) in the required expression. The result we obtain is,

A7'D

A" = ng m((Dk n)71 -+ (Dl1n)nY+

36 g

+(DOPI)^n)- -/"7i( 7"(7(Dsip) + -N(Dvs)

+

ys(Dvou))), (2.5.8) where we have defined Dioj (Di0,-)t7O; and where in the contraction Vi the gamma matrices are to be considered as numerical, not as space dependent. The sum of (2.5.5), (2.5.7) and (2.5.8) according to (2.3.1) finally yields the manifestly gauge invariant Hamiltonian.

As was done for the gluon functional derivative, we shall similarly define a gauge invariant vector-spinor density operator Xi to deal with the gluino functional deriva-tive,

6AHE

)

= g(X ) 1ai(x)x (x).

(2.5.9)

Xi(x) is a

(0)

vector-spinor density (weight

})

under GL(3). With these definitions at hand, one can now express the functional dependence of the wave-functional T[Aa, Ab]

in terms of the new variables. Under a variation, we have,

6T

=d3x {

d

6AZ(x)+

XA

(x)}

=

=

Jd

3

x

{

fg(x)ug

(x)6A"(x)[ei(x)'I

+

g(x)Uai(x)[X4(x)I1]6Aa(x)},

(2.5.10)

where one should use the expression for the variations of the gauge fields in (2.4.17) and (2.4.20-21). Expanding this expression through rather lengthy calculations, it can then be seen that the term in 6u is proportional to the Gauss' law operator (2.3.4), when expressed in geometrical terms, and acting on the wave-functional. Observe

that this is g9aq[g, ] = 0, which in the new variables can be written as,

i Z1 n ik ' n

(ViC 3 + -O7OC3+ 2(gjn il - giOn)(~~n)-ii + (Dli4s)-yn+

2

72g

+(DTn)-s)7 0 7' (k(D, i) + Yi (Dk Or) + yr (D Oi)))F [Y,

4]

= 0. (2.5.11) So, wave-functionals whose dependence is solely on the new gauge invariant variables are gauge invariant, and gauge invariant wave-functionals depend solely on the new gauge invariant variables. It is in these physical gauge invariant wave-functionals that we are mainly interested, and for these the previous expression for 6T reduces to,

6T

[gi,

k]

=

dx {

12"V (gni)[eJ]+

+ n"[ml ]t kOn V(6kki) + +( ng<* iV (6gkl) + 7i 3*kn' (6gkl))+

12

+±nml~f~&b b~n ' w~4i[~I]

4

nm

[Xm'I]7'(iDi(6n)

_{+ YnDi( 60i) +}

-YiD(60n))-6

±nml [X'M IFz'(/ nON0 + <)- &4'k + <)3'/jikk)+

24

+~f~ii(?)fk&?bl+ i/i Vj60~k + i-1160bk) + 7ir nO-kO + 0bl'yjikk +

?/4Yl&6bk))-z nml [XnT17 t(710s(i/ 765) + ?(Si/,l) _+y + 4g.s( 4,₁))} (2.5.12)

12

From here one can now extract expressions for the electric and spinor fields, e 3T and Xi0, in terms of functional derivatives of gauge invariant wave-functionals, with respect to the gauge invariant variables. Observe that for such, one has to solve a linear system of differential equations, therefore involving the inversion of differential operators. One can then conclude that in general both operators e&jI and XiI, will depend non-locally on the functional derivatives 6T/6gig and 6TI/6k. Like in the non-supersymmetric case

[43],

the Hamiltonian will thus be a non-local composite operator.

composite operator that we still would like to express in a manifestly gauge invariant way, i.e., that we would like to geometrize. Such an operator is the supersymmetry generator, (2.3.5-6). In particular, we will look at its structure as depicted in equations (2.3.5), (2.3.25-26), and geometrize the tensor IW'(x). For that, one simply has to make use of the previous formulae into equation (2.3.26), and obtain,

(

0 enmli + I isk(Rnl ik - R_"_k)

e nm + I eg5₃'k (R 13'k - R'" 'k) 0 )

.y'(y(Dr On + _Yn DI/Or + yr DIVOn), _(2.5.13)

from where the supersymmetry generator then follows, according to (2.3.25) and (2.3.5).

Some words are now in order, concerning supersymmetry and its quantum field theoretic representation on the geometrized fields. One of the elements that is present in I' is the non-local operator e&j, thus turning the supersymmetry generator into a non-local composite operator, when expressed in the geometrical variables. As we shall see in the following, information about the Green's functions present in this operator can be obtained, albeit in a formal way. By this, we mean that an explicit construction of these Green's functions can only be obtained given a particular geometrical configuration (see [45] for this same situation in the non-supersymmetric case). Moreover, the geometric supersymmetry generator includes the Riemann tensor which is non-linear in the metric and "gravitino" fields, and their derivatives; one would therefore also prefer to have a geometrical configuration with a high degree of symmetry (a maximal number of Killing vectors), in order to simplify it. An example involving spherical geometries, generalizing the one in

[45]

to this supersymmetric case, shows how this situation could be handled [64].

In the pure Yang-Mills case [43], the calculation of the electric field tensor in-volved the inversion of a differential operator that could generically have zero modes. Subtleties associated to the inversion of such an operator were later handled with the insertion of a deformation into the dreibein defining equation [45]. We shall now see that in this supersymmetric case those problems can be better handled, by computing